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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

An example where topological entropy is continuous


Author: Louis Block
Journal: Trans. Amer. Math. Soc. 231 (1977), 201-213
MSC: Primary 58F15; Secondary 54H20
MathSciNet review: 0461582
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Abstract: Let ent denote topological entropy, and let $ {C^r}({S^1},{S^1})$ denote the space of continuous functions of the circle to itself having r continuous derivatives with the $ {C^r}$ (uniform) topology. Let $ {f_0}$ denote a particular $ {C^2}$ map of the circle ($ {f_0}$ is the first bifurcation point one comes to in a bifurcation from a full three shift to a map with finite nonwandering set). The main results of this paper are the following:

Theorem A. The map ent: $ {C^0}({S^1},{S^1}) \to R \cup \{ \infty \} $ is lower-semicontinuous at $ {f_0}$.

Theorem B. The map ent: $ {C^2}({S^1},{S^1}) \to R$ is continuous at $ {f_0}$.

In proving these two theorems several general results on entropy of mappings of the circle are proved.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0461582-9
Article copyright: © Copyright 1977 American Mathematical Society